Abstract
This experiment aims to investigate the modes of vibration of a trailer rig. With this, a range
of instrumentation is used of varying costs and sophistication. It was discovered that this rig
contained two main modes of vibration – longitudinal and rotational vibration. From the
graphs, it was observed that the damping of the system is approximately linear and is most
likely due to friction damping. It is sufficient to use the LVDT coupled with the digital
oscilloscope for these purposes as it produces accurate results at a good price.
Improvements could be made to the experiment to eliminate sources of error – large part of
it having been human error – and to try to mimic the real life situation more closely.
1.
Introduction
The properties of the road trailer setup will be investigated using a number of
instrumentation techniques with different complexities and costs. The performance of each
of the techniques will be benchmarked while the most appropriate technique will be chosen
and recommended. Furthermore, this report will also investigate how the experimental
arrangement related to normal action of the suspension of the road.
2.
Apparatus
Equipment used to record and analyse the motion:
Equipment
Description
Digital Oscilloscope
To record and analyse the motion of the trailer and show the
results in a graphical manner digitally on screen
7000
XY Plotter
Record the motion of the trailer and show ther esults by plotting
up a graph
Analyses the frequency using a computational method
4000
FFT Analyser
Cost ($)
10000
Table 2.1 – recording the analytical equipment and their costs
Equipment used to transducer the motion of the trailer:
Equipment
Description
LDVT
Transducing the motion of trailer by varying voltages
500
Pointer and Scale
Transducing the motion of the trailer
2.5
Stopwatch
Measure the amount of time per period
10
Table 2.2 – Transducing Equipment and their costs
Cost ($)
3.
Experimental procedure
Longitudinal vibration:
3.1 Method 1: Pointer Scale & Stopwatch
1. Place pointer at the end of the trailer (position1) and adjust the height of the scale to
fit the pointer
2. Excite the system by applying a downward force directly at the centre of gravity of
the trailer.
3. Maintain this excitation and record the number of cycles and time taken to complete
4. Excite the system by applying a singular downward force directly at the centre of
gravity of the trailer.
5. Record the amplitude and period of the vibration (movement of pointer and time by
stopwatch – number of cycles in a certain time).
3.2 Method 2: XY Plotter
1.
2.
3.
4.
Clamp the drive rod at the end of the trailer (position 1) and insert it into the LVDT.
Calibrate the LVDT by measuring a zero point with a multimeter.
Adequately prepare the cy plotter (align paper, turn on servo)
Excite the system by applying downward force directly at the centre of gravity of the
trailer
5. The system’s displacement is recorded via the XY Plotter
3.3 Method 3: Digital oscilloscope
1. Clamp the drive rod at the end of the trailer and insert it into the LVDT.
2. Calibrate the LVDT by measuring a zer0 point with a multimeter.
3. Excite the system by applying downward force directly at the centre of gravity of the
trailer
4. Save the result in a floppy disk
3.4 Method 4: Frequency Analyser (FFT) (Demonstration Only)
1. Clamp the drive rod at the end of the trailer (position 1) and insert it into the LVDT
2. Calibrate the LVDT by measuring a zero point with a multimeter.
3. Excite the system by applying downward force directly at the centre of gravity of the
trailer
4. Record data obtained from the screen
Rotational Vibration:
3.5 Method 2: XY Plotter
1.
2.
3.
4.
Clamp the drive rod at the end of the trailer (position 2) and insert it into the LVDT.
Calibrate the LVDT by measuring a zero point with a multimeter.
Adequately prepare the cy plotter (align paper, turn on servo)
Excite the system by applying a force couple to create a torque which in turn causes
a rocking mode of vibration.
5. The system’s displacement is recorded via the XY Plotter
3.6 Method 3: Digital Oscilloscope
1. Clamp the drive rod at the end of the trailer (position 2) and insert it into the LVDT.
2. Calibrate the LVDT by measuring a zero point with a multimeter.
3. Apply a force couple to create a torque which in turn causes a rocking mode of
vibration.
4. Save the result in a floppy disk
3.7 Method 4: Frequency Analysis (FFT)
This method was not carried out in the experiment due to shortage of time. However, a
demonstration was provided where this method was thoroughly explained.
1. Clamp the drive rod at the end of the trailer (position 3) and insert it into the LVDT
2. Calibrate the LVDT by measuring a zero point with a multimeter.
3. Apply a force couple to create a torque which in turn causes a rocking mode of
vibration.
4. Record data obtained from the screen
Fig 3.1 – Top View of Trailer
Fig 3.2 – Front View of
Trailer (How a Force Couple is Applied)
4.
Results & Calculations
Longitudinal Vibration:
4.1 Method 1: Pointer Scale & Stopwatch
From T1which is attached in the appendix, the following data was obtained:
Average
No. of
Oscillations
7
5
5
5.67
Time (s)
𝑥0 (mm)
1.9
1.8
1.9
1.87
23
26
29
26
Undamped Natural Frequency
𝜔𝑛 = 19.12 rad/s
𝑓𝑛 = 3.04 Hz
Damped Natural Frequency
𝜔𝑑 = 19.09 rad/s
𝑓𝑑 = 3.04 Hz
Logarithmic Decrement
δ = 0.347
Damping Coefficient
𝜉 = 0.055
4.2 Method 2: XY Plotters
From the graph obtained from the plotted (attached in Appendix), the period and
displacements can be measured. The first peak of each group was neglected as it can
be considered to be not of natural vibration. From the T1 (appendix), the damping
coefficient for each set of vibrations was obtained.
Set
1
2
3
4
Average
𝜉
0.029
0.0318
0.0384
0.0195
0.0297
Table 4.2 – Data Values from XY Plotter
4.3 Method 3: Digital Oscilloscope
Two sets of data were collected for the longitudinal vibration. Calculations are made
using the same formulas as in method 2 where the damping ratio can be determined
using the peak values in conjunction with the following equation:
𝑋
𝑙𝑛 𝑋 𝑖
𝑖+1
𝜉=
2
√4𝜋 2 + (ln 𝑋𝑖 )
𝑋𝑖+1
The data obtained from the oscilloscope was then translated into a graph in Microsoft Excel.
Longitudinal Vibration - Set 1
Displacement (mm)
1.5
-2.5
1
0.5
0
-2
-1.5
-1
-0.5
-0.5
0
0.5
1
1.5
2Time (s)2.5
-1
-1.5
Figure 4.3.1 – Data Set 1 for Longitudinal Vibration
Peak 2-3
Peak 3-4
Peak 4-5
δ
0.10
0.12
0.16
𝜉
0.03
0.04
0.05
𝜔𝑑
23.31
25.62
25.38
𝜔𝑛
23.32
25.64
25.41
δ = Logarithmic
Decrement
𝜉 = Damping Coefficient
𝜔𝑑 = Damped Frequency
𝜔𝑛 = Natural Frequency
Table 4.3.1 – Values obtained from Data Set 1
Longitudinal Vibration Set 2
Displacement (mm)
1.5
1
0.5
0
-2.5
-2
-1.5
-1
-0.5
-0.5
0
0.5
1
1.5
2
2.5(s)
Time
-1
Fig 4.3.2 – Data Set 2 for Longitudinal Vibration
Peak 2-3
Peak 3-4
Peak 4-5
𝜉
0.04
0.04
0.06
δ
0.12
0.14
0.19
𝜔𝑑
23.51
25.21
25.96
𝜔𝑛
23.53
25.23
26.01
Table 4.3.2 – Values obtained from Data Set 2
4.4 Method 4: Frequency Analyser (FFT)
Although this experiment was not carried out, a demonstration displaying this was
explained during the laboratory. If values were to be obtained from the display of
the FFT analyser, they can be used to calculate damping coefficient 𝜉. The following
equations would be applied:
𝑄=
Where 𝑓𝑐 = Centre Frequency
𝑓𝑙 = Lower Frequency
𝑓𝑢 = Upper Frequency
𝑓𝑐
𝑓𝑢 − 𝑓𝑙
𝜉=
1
2𝑄
Rotational Vibration:
4.5 Method 2: XY Plotter
From the graph obtained from the plotted (attached in Appendix), the period and
displacements can be measured. The first peak of each group was neglected as it can
be considered to be not of natural vibration. From the T1 (appendix), the damping
coefficient for each set of vibrations was obtained.
𝜉
0.0257
0.0283
0.0269
0.0267
0.0269
Set
1
2
3
4
average
Table 4.5.1 – Damping Coefficient from XY Plot
4.6 Method 3: Digital Oscilloscope
Similarly, the values obtained for the rotational vibration were with the same
formula used in section 4.3. The values obtained were translated into graphs with
Microsoft Excel
Rotational Vibration (Set 1)
0.8
Displacement (mm)
0.6
-2.5
0.4
0.2
-2
-1.5
0
-0.5 -0.2 0
-1
0.5
1
1.5
-0.4
-0.6
-0.8
Fig 4.6.1 – Data Set 1 for Rotational Vibration
Peak 2-3
Peak 3-4
Peak 4-5
δ
0.18
0.27
0.58
𝜉
0.06
0.09
0.18
𝜔𝑑
18.97
19.39
20.35
Table 4.6.1 – Values Obtained from Data Set 1
𝜔𝑛
19.01
19.47
20.68
2
2.5(s)
Time
Rotational Vibration (Set 2)
Displacement (mm)
0.4
0.2
0
-2.5
-2
-1.5
-1
-0.5
-0.2
0
0.5
1
1.5
2
2.5(s)
Time
-0.4
-0.6
Fig 4.6.2 – Data Set 2 for Rotational Vibration
Peak 2-3
Peak 3-4
Peak 4-5
δ
0.15
0.17
0.29
𝜉
0.05
0.06
0.09
𝜔𝑑
18.33
21.91
20.56
𝜔𝑛
18.35
21.95
20.64
Table 4.6.2 – Values Obtained from Data Set 2
5.
Discussion
5.1 Modes of Trailer Vibration
There are two main modes in which the trailer vibrates. The frist is a longitudinal
mode of vibration which involved only a vertical motion of the trailer arms and rod,
also known as a bouncing motion. Another vibration mode is a rocking manner
which involved the rotational motion of the trailer arms relative to the rod.
5.2 Linearity of Damping
Based on the observation of the peaks and the sequential changed of, it can be said
that the damping of the system is approximately linear and mostly at the beginning
of the vibration. While the force and displacement magnitudes are high, the friction
in the springs is the dominant damping source. After this time, when magnitudes are
sufficiently low, the springs are too stiff to deflect (threshold as mentioned later) and
so the tyres take up most of the deflection at which points the curve tends to be
exponential and non-linear (very slightly).
Linearity of peaks
Amplitude
Bounce Set 1
Bounce Set 2
Rotational Set 1
Rotational Set 2
Peaks
Figure 5.2.1 – Linearity of Peaks to demonstrate damping of system
Due to this observation, it can be said that this is friction damping as viscous
damping would give a much more pronounced exponential envelope and less of a
linear curve. Additionally, due to the force required to overcome the friction
threshold, the peaks would just halt in friction damping. Tyres are made of materials
which exhibit hysteresis and are therefore considered non-linear dampers. Another
reason is that the area in contact with the ground changes as a square of the vertical
deflection. Consequently, the force exerted by the tyres on the ground is a quadratic
function of vertical displacement.
5.3 Capability and Cost of Equipment for each Method
1. Pointer Scale & Stopwatch
Cost is very low. However, it required at least two people to collect data; one to
measure the frequency while the others take amplitude readings. This method is
highly inaccurate mostly due to human errors such as reading the amplitudes,
reaction time rate to stopping the stopwatch.
2. XY Plotter
With the cost totalling $4500 (see Table 2.2 for details), LVDT with the XY Plotter
is a relatively accurate method. However, the XY plotter has several limitations.
One such limitation is the slow rate. This is a mechanical property whereby the
movement of the plotter cannot proportionally demonstrate the real amplitude
when the frequency is too high. For example, if the drive rod was oscillation at a
very slow speed, the “pen” would move the whole length of paper. However, as
the frequency increases, the “pen” is unable to cover the whole length within the
short time. It was tested at the frequency of 100Hz, the pen was merely vibrating
on the spot and at 1000Hz, there was no movement. Hence, the XY Plotter
should be sued at low frequencies for optimum results.
3. Digital Oscilloscope
This method is more accurate than the XY plotter as the vibrations are directly
observed as digital data which is then plotted on the screen. It can then be
scrutinized and analysed with more detail as it can be used in computer
programs such as Microsoft Excel and Matlab. These machines cost
approximately $7000 which is almost twice as much as an XY Plotter but is
compensated for by its accuracy.
4. Frequency Analyser (FFT)
This method would produce the most accurate results as it can detect both
lateral and normal vibrations simultaneously and displaying these to the screen.
It’s accuracy comes at a price of $10000. However, only a demonstration was
carried out using this method in order to avoid any accidents as it takes time to
learn its operation.
5.4 Closeness of the Experimental Arrangement Relating to Normal Action of the
Suspension
The experiment’s setup is no really similar to the actual scenario that may happen to
a suspension system on the road. For one, the trailer rig is locked to the ground.
Manually vibrating the rig may lead to inconsistencies as the force applied is not
constant, thus affecting the outcome of the experiment. Moreover, the actual road
may include constant smooth travelling with little vibration or rough terrain that
produces a lot of vibrations or a mixture of both.
5.5 Suggestion for the Types of Equipment
In regards to this experiment, the LDVT and digital oscilloscope is recommended to
be used cooperatively in free vibration analysis. It is adequate in measuring natural
frequency and damping coefficient in this situation. However, if the damping
coefficient accuracy is not an important consideration, the XY plotter will be
sufficient for a big saving.
5.6 Sources of Error
There are several sources of errors. The main factor is due to human error which
includes reading the amplitude from the pointer, reflex timing of the stopwatch,
inconsistent rocking for the torsional vibration and the reading of f c from the FFT
analyser display (it is required we take the centre of the peak different readers may
read different values.) There is also quantisation errors or errors during the A/D
conversion. Most equipment has an acceptable range of error. The values for f u and
fl were required to be exactly 3db on either side of fc. However, due to the allowable
“ruler” limitation, our values were taken at approximately 2.8 and 2.7db on either
side. This may lead us to produce slightly different values of 𝜉.
.
The non-linearity due to friction damping may contribute to errors. The limitation of
servo frequency response may not allow equipment to cope with certain levels of
amplitudes. The “clipped” peak of the first peaks produced by the digital oscilloscope
is due to the range allowed - this can be adjusted. There is also the presence of
noise; other natural frequency sources within the system. This is shown by the
jagged curve from the digital oscilloscope. For the rotational vibration, it must be
noted that to obtain a more accurate curve, it is noted that the position of the drive
rod is important to obtain a more accurate curve. Additionally, the method in which
the system may be “rocked” may cause slight inaccuracy due to physical human
error.
5.7 Improvement ideas
Some improvements could be made while running the experiment. High amplitudes
and frequencies should be avoided; measurements should be taken digitally to avoid
human error. If using the digital oscilloscope, a low pass filter could be added to
reduce noise. If taking the measurements of rotational vibration, it is best if the LVDT
w as attached further away from the rig so the displacement it experiences is as
accurate as possible. However, this may prove difficult as the amplitude would now
increase and the LVDT’s range may not be able to cope.
6 Conclusion
From experiments, this set up has two modes of vibration; rotational and
longitudinal which are both approximately linear. Comparatively, it is not necessary
to use highly sophisticated equipment which is much more costly than the LVDT to
obtain our evaluation. Hence the most ideal method which takes cost into
consideration is to use the digital oscilloscope accompanied by the LVDT. Although
the results were fairly good, there were many sources of error, the largest due to
human error. Improvements to this experiment could be made to decrease number
of errors or try to simulate more closely the realness of the situation.